I am reading the book geometry of schemes by Eisenbud and Harris. I am having trouble verifying some statements of Example IV-17, concerning the blow-up of affine space in a point. I am fairly familiar with the blow-up in the language of varieties. However, I do not have much experience with schemes.

The setting is as follows. Consider the affine plane Spec $A[x_1,...,x_n]$ over some ring $A$. Define $T_i:=A\left[\frac{x_1}{x_i},\dots,\frac{x_n}{x_i},x_i\right]$ and $U_i := $Spec$T_i$.

Compute $$ (T_i)_{x_j}=A\left[\frac{x_1}{x_i},\dots,\frac{x_n}{x_i},x_i,\frac{1}{x_j}\right], $$ $$ (T_j)_{x_i}=A\left[\frac{x_1}{x_j},\dots,\frac{x_n}{x_j},x_j,\frac{1}{x_i}\right]. $$ It is clear that we can identify these rings in $A\left[x_1,...,x_n,\frac{1}{x_1},\dots,\frac{1}{x_n}\right]$. This identification yields isomorphisms of schemes between $(U_i)_{x_j}$ and $(U_j)_{x_i}$. Then, the author defines the blow-up $Z$ to be the gluing of the $U_i$'s via these isomorphisms. The map that comes with this blow-up is the gluing of the maps $U_i\rightarrow$ Spec $A[x_1,...,x_n]$ associated to the ring morphisms

\begin{matrix} A[x_1,...,x_n] & \hookrightarrow & T_i\\ x_i & \mapsto & x_i\\ x_j & \mapsto & \frac{x_j}{x_i}x_i. \end{matrix}

This gives a map $\phi:Z\rightarrow $Spec $A[x_1,...,x_n]$. I am able to figure out that the exceptional divisor (the inverse image of the origin under this map $\phi$) is isomorphic to $\mathbb{P}^{n-1}$. However, I don't see how to show that $\phi$ is an isomorphism outside this exceptional divisor.

Does anyone know how to do this? Thank you in advance.


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